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Events for November 2012 – March 2013

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Nov. 06, 2012 2pm (MATH 350)	Lie Theory Justin Keller (CU) X We define the notion of a supercharacter theory (SCT) for an arbitrary finite group and view each SCT as a coarsening or approximation of the usual character theory of the group. Generalizing the relationship between the center of the group algebra and its character theory, we describe how certain subalgebras (Schur rings) of the group algebra play an analogous role for SCTs. In this way, Schur rings can be viewed as an approximate center for the group algebra, and as a tool for finding and understanding SCTs. The goal will be to provide a foundation and motivating example (the group algebra) for the following talk, which will introduce the notion of a Schur ring in the setting of Hopf algebras. Supercharacter Theories and Schur Rings II
Nov. 07, 2012 4:15pm (MATH 3…	Grad Student Seminar Trubee Davison X Given a complete and separable metric space (X,d), and a finite family, S, of contractions on X, one can construct a compact subset K of X that is invariant under S. This set K is called the fractal set, or attractor set, associated to S. In this talk, we will discuss the construction of the fractal set. As a somewhat loose, but fun, application of fractals, we will then look at one example of the 'fractal-type' phrasing that occurs in many of Bach's compositions. The speaker will conclude with a performance on the violin of a Bach composition. Constructing the Fractal Set with an Application to the Music of Bach
Nov. 13, 2012 2pm (MATH 350)	Lie Theory Scott Andrews (CU) X Classifying the irreducible characters of the group of unipotent upper triangular matrices over a finite field is a provably difficult problem. We can instead consider a coarsening of the usual character theory, which we will refer to as a supercharacter theory. I will construct a supercharacter theory of the group of unipotent upper triangular matrices and look at analogous constructions in other types. Supercharacter Theories of Unipotent Groups
Nov. 27, 2012 2pm (MATH 350)	Lie Theory Aaron Lauda (USC) X It is a well understood story that one can extract link invariants from quantum deformations of the universal enveloping algebra of simple Lie algebra called quantum groups. These invariants are called Reshetikhin-Turaev invariants and the famous Jones polynomial is the simplest example. Kauffman showed that the Jones polynomial could be described very simply by replacing crossings in a knot diagrams by various smoothings. In this talk we will explain Cautis-Kamnitzer-Licata's simple new approach to understanding these invariants using basic representation theory and the quantum Weyl group action based on the representation theoretic notion called skew-Howe duality. Even the graphical (or skein theory) description of these invariants can be recovered in an elementary way from this data. The advantage of this approach is that it suggests a `categorification' where knot homology theories arise in an elementary way from the structure of categorified quantum groups. Why representation theorists already know about quantum knot invariants Sponsored by the Meyer Fund
Nov. 28, 2012 4:15pm (MATH 3…	Grad Student Seminar Ryan Rosenbaum X A transcendental number is a complex number which is not the root of a polynomial in ℚ[x]. We will give an accessible introduction to the theory of real transcendental numbers. We will touch on basic properties of transcendental numbers (like they actually exist), Liouville's theorem, irrationality measure and Roth's theorem. Time permitting we will include a construction of transcendental numbers relating to the undecidability of the halting problem over Turing machines. An Introduction to Transcendental Number Theory
Dec. 11, 2012 2pm (MATH 350)	Lie Theory Vivek Dhand (MSU) X Young's partition lattice L(m,n) consists of unordered partitions having m parts where each part is at most n, ordered by inclusion of the corresponding Young diagrams. Many years ago, Richard Stanley conjectured that this poset has a decomposition into saturated rank-symmetric chains. Despite many efforts, this conjecture has only been proved when min(m,n) is less than or equal to 4. In this talk, we decompose L(m,n) into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we show that resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of L(m,n) consisting of "sufficiently generic" partitions. Tropical Decomposition of Young's Partition Lattice
Jan. 30, 2013 4:15pm (MATH 3…	Grad Student Seminar Nathan Wakefield X Heights are an important component of the number theorist's toolbox. This presentation will describe the basic elements of heights, and how they play a role in many major areas of number theory; ranging from elliptic curves to dynamical systems. The canonical height is a particularly important tool in arithmetic dynamics. In this talk, we will discuss recent developments in the area of canonical heights, and explain an extension to generalized iteration. We will conclude with an open conjecture and discuss how heights may contribute to its solution. Canonical Heights in Generalized Iterations
Feb. 05, 2013 2pm (MATH 350)	Lie Theory Steve Doty (Loyola) X Generalized Schur algebras were introduced in the late 1980s by S. Donkin. They are one way of extending the classical theory from Type A root systems to arbitrary type, and of course q-analogues exist. In 2002 Giaquinto and I found two presentations of the classical Schur algebras and their q-analogues; one of them generalized to arbitrary type. One of the most important facts about generalized q-Schur algebras is that they are quasihereditary (in the sense of Cline-Parshall-Scott). I proved this in 2003 as an application of known facts about the Lusztig-Kashiwara canonical-crystal bases for quantized enveloping algebras of semisimple Lie algebras. Recently, Giaquinto and I found an elementary proof that does not depend on the deep existence results of Lusztig and Kashiwara. Instead, the proof starts from the defining presentation and uses only known results on Lie algebras and elementary calculations. Generalized (q)-Schur algebras and their quasihereditary structure I
Feb. 12, 2013 2pm (MATH 350)	Lie Theory Steve Doty (Loyola) X Generalized Schur algebras were introduced in the late 1980s by S. Donkin. They are one way of extending the classical theory from Type A root systems to arbitrary type, and of course q-analogues exist. In 2002 Giaquinto and I found two presentations of the classical Schur algebras and their q-analogues; one of them generalized to arbitrary type. One of the most important facts about generalized q-Schur algebras is that they are quasihereditary (in the sense of Cline-Parshall-Scott). I proved this in 2003 as an application of known facts about the Lusztig-Kashiwara canonical-crystal bases for quantized enveloping algebras of semisimple Lie algebras. Recently, Giaquinto and I found an elementary proof that does not depend on the deep existence results of Lusztig and Kashiwara. Instead, the proof starts from the defining presentation and uses only known results on Lie algebras and elementary calculations. Generalized (q)-Schur algebras and their quasihereditary structure II
Feb. 13, 2013 4:15pm (MATH 3…	Grad Student Seminar Amy Feaver X The Mahler measure of a complex-valued polynomial P(z) is the geometric mean of the modulus of the polynomial evaluated over the unit circle. At first glance, this quantity appears to be a harmless integral, but it turns out that it invades several areas of number theory. In this talk, we will discuss the relationship between Mahler measure, the golden ratio and algebraic integers. We will also extend the definition of Mahler measure to multivariable functions, and will conclude with evaluating the Mahler measure of a polynomial in three variables, finding that it can be expressed in terms of the Riemann zeta function evaluated at 3. Mahler measure
Feb. 26, 2013 2pm (MATH 350)	Lie Theory Shawn Burkett (CU) X Motivated by the problem of bounding the order of a group with a large character degree, Martin Isaacs predicted that the largest degree among irreducible complex representations of a finite nonabelian simple group could be bounded in terms of the smaller degrees. Using the Jordan decomposition of characters, we answer this question in the affirmative for a few infinite families of finite simple classical groups. We also discuss a recent theorem of Michael Larson, Gunter Malle and Pham Tiep regarding the largest irreducible representations of simple groups, as well as Isaacs's application of this theorem to the problem of bounding the orders of certain groups. Bounding the Largest Irreducible Character Degree of a Finite Simple Group
Feb. 27, 2013 4:15pm (MATH 3…	Grad Student Seminar Bryce Chriestenson X This talk will present certain homological invariants attached to a stratified space \(X\), called the Whitney-deRham cohomology. This cohomology is defined as the cohomology of a chain complex, \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\), associated to \(X\) in a somewhat ad-hoc way. The main result is to show that though the definition of \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) depends on several choices, when certain conditions are imposed on \(X\), the Whitney-deRham cohomology only depends on the homotopy type of \(X\). This is achieved by showing that \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) can be realized as a fine complex of sheaves which is a resolution of the locally constant sheaf on \(X\). An application of this work is in the area of homotopy theory. One can canonically define a commutative differential graded algebra(CDGA), \(A_{PL}\left(X\right)\), on \(X\) in such a way that any CDGA which is quasi-isomorphic to it determines the real homotopy type of \(X\). It will be shown that the complex \(\Omega^{ * }_{\mathrm{ W } } \left( X \right)\) is quasi-isomorphic to \(A_{PL}\left(X\right)\), and thus determines the real homotopy type of \(X\). The real homotopy type of singular spaces via the Whitney-deRham Complex
Mar. 19, 2013 2pm (MATH 350)	Lie Theory Chris Bowman (Paris 7) X We discuss the partition algebra as a setting for the study of row-removal phenomena for the symmetric group. We describe its representation theory in terms of a type A facet geometry. We then discuss applications to the Kronecker problem, in particular a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition. The representation theory of the partition algebra
Mar. 20, 2013 4:15pm (MATH 3…	Grad Student Seminar Matthew Grimes X Answering global questions with local data is a common theme in geometry, and sheaves are a powerful tool for organizing such problems. Cech cohomology aids in the study of sheaves on topological spaces, and helps us understand when local data is enough to answer a global question. We will discuss the basics of sheaves, and then go over the construction of Cech cohomology. We will examine basic theorems with an emphasis on motivation, examples, and accessibility. Applications to topology and geometry will also be covered, and so a cursory review of simplicial homology and complex analysis prior to the talk may be useful. Sheaves and Cech Cohomology

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